As discussion in the seminal paper [1]}, one could consider a class of extremal black hole, which have geometry of \(AdS_2 \times S^{D-2}\)  in the near horizon limit. One can define the entropy function
via the Legendre transformation of the integral for the Lagrangian density evaluated for this black hole background over the sphere. Note that, as pointed out in [1]}, although the original discussion in [3]} relies on the geometry of \(AdS_2 \times S^{D-2}\) , it can be valid for the planar AdS black hole horizon with geometry \(AdS_2 \times R^{D-2}\) .
